Answer
**a) Value of \( m \):**
\[
m = \frac{1}{2}
\]
**b) Sequence:**
\[
\frac{1}{2}, \quad 3, \quad \frac{11}{2}
\]
or
\[
0.5, \quad 3, \quad 5.5
\]
with a common difference of \( \frac{5}{2} \) (2.5).
Solution
Let's solve the given problem step by step.
**Given:**
Three consecutive terms of an arithmetic sequence are:
- First term: \( m \)
- Second term: \( 2m + 2 \)
- Third term: \( 5m + 3 \)
An **arithmetic sequence** is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is known as the **common difference** (\( d \)).
### a) Determine the value of \( m \)
To find the value of \( m \), we'll set up an equation based on the property of arithmetic sequences.
1. **Find the common difference between the first and second terms:**
\[
d = (2m + 2) - m = m + 2
\]
2. **Find the common difference between the second and third terms:**
\[
d = (5m + 3) - (2m + 2) = 3m + 1
\]
3. **Since the common difference is the same for both pairs of consecutive terms, set them equal:**
\[
m + 2 = 3m + 1
\]
4. **Solve for \( m \):**
\[
m + 2 = 3m + 1 \\
2 - 1 = 3m - m \\
1 = 2m \\
m = \frac{1}{2}
\]
**Answer to part (a):**
\[
m = \frac{1}{2}
\]
### b) Determine the sequence
Now that we have the value of \( m \), we can find the three consecutive terms of the arithmetic sequence.
1. **First term (\( a_1 \)):**
\[
a_1 = m = \frac{1}{2}
\]
2. **Second term (\( a_2 \)):**
\[
a_2 = 2m + 2 = 2 \left( \frac{1}{2} \right) + 2 = 1 + 2 = 3
\]
3. **Third term (\( a_3 \)):**
\[
a_3 = 5m + 3 = 5 \left( \frac{1}{2} \right) + 3 = \frac{5}{2} + 3 = \frac{11}{2} = 5.5
\]
4. **Common difference (\( d \)):**
\[
d = m + 2 = \frac{1}{2} + 2 = \frac{5}{2} = 2.5
\]
5. **General form of the sequence:**
\[
a_n = a_1 + (n-1)d
\]
Plugging in the known values:
\[
a_n = \frac{1}{2} + (n-1) \times \frac{5}{2}
\]
**Answer to part (b):**
The three consecutive terms of the arithmetic sequence are:
\[
\frac{1}{2}, \quad 3, \quad \frac{11}{2}
\]
or equivalently:
\[
0.5,\quad 3,\quad 5.5
\]
The common difference is \( \frac{5}{2} \) (or 2.5), and the general formula for the \( n \)-th term of the sequence is:
\[
a_n = \frac{1}{2} + (n-1) \times \frac{5}{2}
\]
Reviewed and approved by the UpStudy tutoring team
Explain
Simplify this solution
